# Order of integration and differentiation?

1. Jun 9, 2007

### Repetit

What am I doing wrong here, I thought the order of integration and differentiation didn't matter in most cases:

$$\int_a^b \frac{d}{dx} f(x) dx = \frac{d}{dx} \int_a^b f(x) dx = \frac{d}{dx} (F(b) - F(a)) = 0$$

This is zero no matter what the expression of f(x) because F(a) and F(b) are constants. Am I not allowed to take the differentiation outside the integral when the integral has limits? This is most likely a stupid question for reasons I cannot see.

2. Jun 9, 2007

### cepheid

Staff Emeritus
$$\int_a^b f(x) \, dx$$

is just a number, so your result shouldn't surprise you.

$$\int_a^b \frac{d}{dx}f(x) \, dx$$

is a (most probably different) number. So I guess the answer to your question is:

$$\int_a^b \frac{d}{dx}f(x) \, dx \not= \frac{d}{dx}\int_a^b f(x) \, dx$$

3. Jun 9, 2007

### Hurkyl

Staff Emeritus
I would like to point out that the three uses of the letter x in

$$\frac{d}{dx} \int_a^b f(x) \, dx$$

cannot all possibly refer to the same thing. You have written the mathematical equivalent of either gibberish or equivocation.

gibberish, meaning that it's not a well-formed mathematical expression

equivocation, meaning that you are using the same name for different things, and subsequently treating them as if they were the same thing.

4. Jun 10, 2007

### cepheid

Staff Emeritus
Can you elaborate on this? I mean, it seems like the two inside the integration sign are fine. The function depends on x. It is being integrated with respect to x. And it's just a definite integral...a number. Why can't it be differentiated wrt x as well?

If it were something like this:

$$\frac{d}{dx} \int_a^x f(x) \, dx = f(x)$$

then I could see why you'd complain. Some dummy variable of integration should be used:

$$\frac{d}{dx} \int_a^x f(t) \, dt = f(x)$$

Last edited: Jun 10, 2007
5. Jun 10, 2007

### ice109

$$f(x)=5x^2$$
$$\int_a^b f(x) \, dx = \int_a^b (5x^2) \, dx = \frac{5}{3}b^3 - \frac{5}{3}a^3$$

i don't understand how any of that is wrong except for the last part where differentiating a definite integral, which is a number, yields anything but 0

6. Jun 10, 2007

### DeadWolfe

Hurkyl is complaining about the use of the symbol x to denote a dummy variable and a non-dummy variable.

7. Jun 10, 2007

### ice109

i don't understand what that means?

8. Jun 10, 2007

### cepheid

Staff Emeritus
That's just point of what I was asking! I don't think there *is* a need for a dummy variable in this situation. Didn't anybody read my post #4? I even gave an example of a situation in which you *would* need a dummy variable.

9. Jun 10, 2007

### ice109

yEA im with you!

10. Jun 10, 2007

### D H

Staff Emeritus
In a definite integral, the symbol dx (or dwhatever) denotes that "x" is a "dummy" variable of integration. The integration variable vanishes once the function is integrated and the integration limits are applied. As Hurkl noted, using the same variable as the variable of integration and outside the integral leads to gibberish. The possibility for confusion become even less greater if the integration limits are functions rather than constants. For this reason, it is preferable to use a notation like cepheid did in post #4. Using a distinct variable of integration becomes mandatory when you do things like

$$\frac{d}{dt} \int_{a(t)}^{b(t)} f(t, \tau)\,\mathrm{d}\tau$$

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